What is the integral of (sec(x))/(asec(x)-acos(x)) ?

The integral of (sec(x))/(asec(x)-acos(x)) is -1/a cot(x)+C

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integral of (sec(x))/(asec(x)-acos(x))

\int \frac{sec ( x )}{a sec ( x ) - a cos ( x )} d x

- \frac{1}{a} cot (x) + C

Solution steps

\int \frac{sec ( x )}{a sec ( x ) - a cos ( x )} d x

\frac{sec ( x )}{a sec ( x ) - a cos ( x )} = \frac{1}{a} \frac{sec ( x )}{sec ( x ) - cos ( x )}

= \int \frac{1}{a} \cdot \frac{sec ( x )}{sec ( x ) - cos ( x )} d x

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= \frac{1}{a} \cdot \int \frac{sec ( x )}{sec ( x ) - cos ( x )} d x

Rewrite using trig identities

= \frac{1}{a} \cdot \int \frac{1}{1 - cos ^{2} ( x )} d x

Rewrite using trig identities

= \frac{1}{a} \cdot \int csc^{2} (x) d x

Use the common integral:

\int csc^{2} (x) d x = - cot (x)

= \frac{1}{a} (- cot (x))

Simplify

= - \frac{1}{a} cot (x)

Add a constant to the solution

= - \frac{1}{a} cot (x) + C

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What is the integral of (sec(x))/(asec(x)-acos(x)) ?
The integral of (sec(x))/(asec(x)-acos(x)) is -1/a cot(x)+C